Empirical likelihood ratio confidence intervals for a single functional. (English) Zbl 0641.62032
Let \((X_ 1,...,X_ n)\) be a random sample, its components \(X_ i\) are observations from a distribution-function \(F_ 0\). The empirical distribution function \(F_ n\) is a nonparametric maximum likelihood estimate of \(F_ 0\). \(F_ n\) maximizes
\[
L(F)=\prod^{n}_{i=1}\{F(X_ i)-F(X_ i-)\}
\]
over all distribution functions F. Let \(R(F)=L(F)/L(F_ n)\) be the empirical likelihood ratio function and T(.) any functional. It is shown that sets of the form
\[
\{T(F)| R(F)\geq c\}
\]
may be used as confidence regions for some \(T(F_ 0)\) like the sample mean or a class of M-estimators (especially the quantiles of \(F_ 0)\). These confidence intervals are compared in a simulation study to some bootstrap confidence intervals and to confidence intervals based on a t-statistic for a confidence coefficient \(1-\alpha =0.9\). It seems that two of the bootstrap intervals may be recommended but the simulation is based on 1000 runs only.
Reviewer: D. Rasch
MSC:
62G15 | Nonparametric tolerance and confidence regions |
62G30 | Order statistics; empirical distribution functions |
62G05 | Nonparametric estimation |